On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle

ABSTRACT

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

In this work, we used the complex variable methods to derive the Goursat functions for the first and second fundamental problem of an infinite plate with a curvilinear hole C. The hole is mapped in the domain inside a unit circle by means of the rational mapping function. Many special cases are discussed and established of these functions. Also, many applications and examples are considered. The results indicate that the infinite plate with a curvilinear hole inside the unit circle is very pronounced.

KEYWORDS

Complex Variable Method, An Infinite Plate, Curvilinear Hole, Conformal Mapping, Goursat Functions

Complex Variable Method, An Infinite Plate, Curvilinear Hole, Conformal Mapping, Goursat Functions

Cite this paper

Bayones, F. and Alharbi, B. (2015) On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle.*Applied Mathematics*, **6**, 206-220. doi: 10.4236/am.2015.61020.

Bayones, F. and Alharbi, B. (2015) On a Problem of an Infinite Plate with a Curvilinear Hole inside the Unit Circle.

References

[1] Muskhelishvili, N.I. (1953) Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland.

[2] Spiegel. M.R. (1964) Theory and Problems of Complex Variables, Schaum’s Outline Series. McGraw-Hill, New York.

[3] Rubenfeld, L.A. (1985) A First Course in Applied Complex Variables. John Wiley & Sons, New York.

[4] Bieberbach, L. (1953) Conformal Mapping. Chelsea Publishing Company, New York.

[5] Popov, G.Ya. (1982) Contact Problems for a Linearly Deformable Functions. Odessa, Kiev.

[6] Sabbah, A.S., Abdou, M.A. and Ismail, A.S. (2002) An Infinite Plate with a Curvilinear Hole and Flowing Heat. Proc. Math. Phys. Soc. Egypt.

[7] Atkin, R.J. and Fox, N. (1990) An Introduction to the Theory of Elasticity. Longman, Harlow.

[8] Colton, D. and Kress, R. (1983) Integral Equation Method in Scattering Theory. John Wiley, New York.

[9] Abdou, M.A. (2003) On Asymptotic Method for Fredholm-Volterra Integral Equation of the Second Kind in Contact Problems. Journal of Computational and Applied Mathematics, 154, 431-446.

http://dx.doi.org/10.1016/S0377-0427(02)00862-2

[10] Noda, N. and Hetnarski Yoshinobu Tanigowa, R.B. (2003) Thermal Stresses. Taylor and Francis, UK.

[11] Hetnarski, R.B. (2004) Mathematical Theory of Elasticity. Taylor and Francis, London.

[12] Parkus, H. (1976) Thermoelasticity. Springer-Verlag, New York.

[13] Gakhov, F.D. (1966) Boundary Value Problems. General Publishing Company, Ltd., Canada

[14] Ciarlet, P.G., Schultz, M.H. and Varga, R.S. (1967) Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problems. I. One Dimensional Problem. Numerische Mathematik, 9, 394-430.

[15] Zebib, A. (1984) A Chebyshev Method for the Solution of Boundary Value Problems. Journal of Computational Physics, 53, 443-455.

http://dx.doi.org/10.1016/0021-9991(84)90070-6

[16] Saito, S. and Yamamto, M. (1989) Boundary Value Problems of Quasilinear Ordinary Differential Systems on a Finite Interval. Math. Japon., 34, 447-458.

[17] Abdou, M.A. (1994) First and Second Fundamental Problems for an Elastic Infinite Plate with a Curvilinear Hole. Alex. Eng. J., 33, 227-233.

[18] Abdou, M.A. (2002) Fundamental Problems for an Infinite Plate with a Curvilinear Hole Having Infinite Poles. Applied Mathematics and Computation, 125, 177-193.

[19] Abdou, M.A. and Khamis, A.K. (2000) On a Problem of an Infinite Plate with a Curvilinear Hole Having Three Poles and Arbitrary Shape. Bulletin of the Calcutta Mathematical Society, 92, 309-322.

[20] Abd-Alla, A.M., Abo-Dahab, S.M. and Bayones, F.S. (2013) Propagation of Rayleigh Waves in Magneto-Thermo-Elastic Half-Space of a Homogeneous Orthotropic Material under the Effect of Rotation, Initial Stress and Gravity Field. Journal of Vibration and Control, 19, 1395-1420.

http://dx.doi.org/10.1177/1077546312444912

[21] Abd-Alla, A.M. and Abo-Dahab, S.M. (2012) Effect of Rotation and Initial Stress on an Infinite Generalized Magneto-Thermoelastic Diffusion Body with a Spherical Cavity. Journal of Thermal Stresses, 35, 892-912.

http://dx.doi.org/10.1080/01495739.2012.720209

[22] Abd-Alla, A.M., Abd-Alla, A.N. and Zeidan, N.A. (2000) Thermal Stresses in a Non-Homogeneous Orthotropic Elastic Multilayered Cylinder. Journal of Thermal Stresses, 23, 413-428.

[23] Abd-Alla, A.M., Mahmoud, S.R., Abo-Dahab, S.M. and Helmy, M.I. (2011) Propagation of S-Wave in a Non-Homo-geneous Anisotropic Incompressible and Initially Stressed Medium under Influence of Gravity Field. Applied Mathematics and Computation, 217, 4321-4332.

http://dx.doi.org/10.1016/j.amc.2010.10.029

[24] Abd-Alla, A.M., Mahmoud, S.R. and AL-Shehri, N.A. (2011) Effect of the Rotation on a Non-Homogeneous Infinite Cylinder of Orthotropic Material. Applied Mathematics and Computation, 217, 8914-8922.

[1] Muskhelishvili, N.I. (1953) Some Basic Problems of Mathematical Theory of Elasticity. Noordroof, Holland.

[2] Spiegel. M.R. (1964) Theory and Problems of Complex Variables, Schaum’s Outline Series. McGraw-Hill, New York.

[3] Rubenfeld, L.A. (1985) A First Course in Applied Complex Variables. John Wiley & Sons, New York.

[4] Bieberbach, L. (1953) Conformal Mapping. Chelsea Publishing Company, New York.

[5] Popov, G.Ya. (1982) Contact Problems for a Linearly Deformable Functions. Odessa, Kiev.

[6] Sabbah, A.S., Abdou, M.A. and Ismail, A.S. (2002) An Infinite Plate with a Curvilinear Hole and Flowing Heat. Proc. Math. Phys. Soc. Egypt.

[7] Atkin, R.J. and Fox, N. (1990) An Introduction to the Theory of Elasticity. Longman, Harlow.

[8] Colton, D. and Kress, R. (1983) Integral Equation Method in Scattering Theory. John Wiley, New York.

[9] Abdou, M.A. (2003) On Asymptotic Method for Fredholm-Volterra Integral Equation of the Second Kind in Contact Problems. Journal of Computational and Applied Mathematics, 154, 431-446.

http://dx.doi.org/10.1016/S0377-0427(02)00862-2

[10] Noda, N. and Hetnarski Yoshinobu Tanigowa, R.B. (2003) Thermal Stresses. Taylor and Francis, UK.

[11] Hetnarski, R.B. (2004) Mathematical Theory of Elasticity. Taylor and Francis, London.

[12] Parkus, H. (1976) Thermoelasticity. Springer-Verlag, New York.

[13] Gakhov, F.D. (1966) Boundary Value Problems. General Publishing Company, Ltd., Canada

[14] Ciarlet, P.G., Schultz, M.H. and Varga, R.S. (1967) Numerical Methods of High-Order Accuracy for Nonlinear Boundary Value Problems. I. One Dimensional Problem. Numerische Mathematik, 9, 394-430.

[15] Zebib, A. (1984) A Chebyshev Method for the Solution of Boundary Value Problems. Journal of Computational Physics, 53, 443-455.

http://dx.doi.org/10.1016/0021-9991(84)90070-6

[16] Saito, S. and Yamamto, M. (1989) Boundary Value Problems of Quasilinear Ordinary Differential Systems on a Finite Interval. Math. Japon., 34, 447-458.

[17] Abdou, M.A. (1994) First and Second Fundamental Problems for an Elastic Infinite Plate with a Curvilinear Hole. Alex. Eng. J., 33, 227-233.

[18] Abdou, M.A. (2002) Fundamental Problems for an Infinite Plate with a Curvilinear Hole Having Infinite Poles. Applied Mathematics and Computation, 125, 177-193.

[19] Abdou, M.A. and Khamis, A.K. (2000) On a Problem of an Infinite Plate with a Curvilinear Hole Having Three Poles and Arbitrary Shape. Bulletin of the Calcutta Mathematical Society, 92, 309-322.

[20] Abd-Alla, A.M., Abo-Dahab, S.M. and Bayones, F.S. (2013) Propagation of Rayleigh Waves in Magneto-Thermo-Elastic Half-Space of a Homogeneous Orthotropic Material under the Effect of Rotation, Initial Stress and Gravity Field. Journal of Vibration and Control, 19, 1395-1420.

http://dx.doi.org/10.1177/1077546312444912

[21] Abd-Alla, A.M. and Abo-Dahab, S.M. (2012) Effect of Rotation and Initial Stress on an Infinite Generalized Magneto-Thermoelastic Diffusion Body with a Spherical Cavity. Journal of Thermal Stresses, 35, 892-912.

http://dx.doi.org/10.1080/01495739.2012.720209

[22] Abd-Alla, A.M., Abd-Alla, A.N. and Zeidan, N.A. (2000) Thermal Stresses in a Non-Homogeneous Orthotropic Elastic Multilayered Cylinder. Journal of Thermal Stresses, 23, 413-428.

[23] Abd-Alla, A.M., Mahmoud, S.R., Abo-Dahab, S.M. and Helmy, M.I. (2011) Propagation of S-Wave in a Non-Homo-geneous Anisotropic Incompressible and Initially Stressed Medium under Influence of Gravity Field. Applied Mathematics and Computation, 217, 4321-4332.

http://dx.doi.org/10.1016/j.amc.2010.10.029

[24] Abd-Alla, A.M., Mahmoud, S.R. and AL-Shehri, N.A. (2011) Effect of the Rotation on a Non-Homogeneous Infinite Cylinder of Orthotropic Material. Applied Mathematics and Computation, 217, 8914-8922.